// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2006-2009 Benoit Jacob <jacob.benoit.1@gmail.com>
//
// This Source Code Form is subject to the terms of the Mozilla
// Public License v. 2.0. If a copy of the MPL was not distributed
// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.

#ifndef EIGEN_LU_H
#define EIGEN_LU_H

namespace Eigen {

namespace internal {
template<typename _MatrixType>
struct traits<FullPivLU<_MatrixType>> : traits<_MatrixType>
{
	typedef MatrixXpr XprKind;
	typedef SolverStorage StorageKind;
	typedef int StorageIndex;
	enum
	{
		Flags = 0
	};
};

} // end namespace internal

/** \ingroup LU_Module
 *
 * \class FullPivLU
 *
 * \brief LU decomposition of a matrix with complete pivoting, and related features
 *
 * \tparam _MatrixType the type of the matrix of which we are computing the LU decomposition
 *
 * This class represents a LU decomposition of any matrix, with complete pivoting: the matrix A is
 * decomposed as \f$ A = P^{-1} L U Q^{-1} \f$ where L is unit-lower-triangular, U is
 * upper-triangular, and P and Q are permutation matrices. This is a rank-revealing LU
 * decomposition. The eigenvalues (diagonal coefficients) of U are sorted in such a way that any
 * zeros are at the end.
 *
 * This decomposition provides the generic approach to solving systems of linear equations, computing
 * the rank, invertibility, inverse, kernel, and determinant.
 *
 * This LU decomposition is very stable and well tested with large matrices. However there are use cases where the SVD
 * decomposition is inherently more stable and/or flexible. For example, when computing the kernel of a matrix,
 * working with the SVD allows to select the smallest singular values of the matrix, something that
 * the LU decomposition doesn't see.
 *
 * The data of the LU decomposition can be directly accessed through the methods matrixLU(),
 * permutationP(), permutationQ().
 *
 * As an example, here is how the original matrix can be retrieved:
 * \include class_FullPivLU.cpp
 * Output: \verbinclude class_FullPivLU.out
 *
 * This class supports the \link InplaceDecomposition inplace decomposition \endlink mechanism.
 *
 * \sa MatrixBase::fullPivLu(), MatrixBase::determinant(), MatrixBase::inverse()
 */
template<typename _MatrixType>
class FullPivLU : public SolverBase<FullPivLU<_MatrixType>>
{
  public:
	typedef _MatrixType MatrixType;
	typedef SolverBase<FullPivLU> Base;
	friend class SolverBase<FullPivLU>;

	EIGEN_GENERIC_PUBLIC_INTERFACE(FullPivLU)
	enum
	{
		MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
		MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime
	};
	typedef typename internal::plain_row_type<MatrixType, StorageIndex>::type IntRowVectorType;
	typedef typename internal::plain_col_type<MatrixType, StorageIndex>::type IntColVectorType;
	typedef PermutationMatrix<ColsAtCompileTime, MaxColsAtCompileTime> PermutationQType;
	typedef PermutationMatrix<RowsAtCompileTime, MaxRowsAtCompileTime> PermutationPType;
	typedef typename MatrixType::PlainObject PlainObject;

	/**
	 * \brief Default Constructor.
	 *
	 * The default constructor is useful in cases in which the user intends to
	 * perform decompositions via LU::compute(const MatrixType&).
	 */
	FullPivLU();

	/** \brief Default Constructor with memory preallocation
	 *
	 * Like the default constructor but with preallocation of the internal data
	 * according to the specified problem \a size.
	 * \sa FullPivLU()
	 */
	FullPivLU(Index rows, Index cols);

	/** Constructor.
	 *
	 * \param matrix the matrix of which to compute the LU decomposition.
	 *               It is required to be nonzero.
	 */
	template<typename InputType>
	explicit FullPivLU(const EigenBase<InputType>& matrix);

	/** \brief Constructs a LU factorization from a given matrix
	 *
	 * This overloaded constructor is provided for \link InplaceDecomposition inplace decomposition \endlink when \c
	 * MatrixType is a Eigen::Ref.
	 *
	 * \sa FullPivLU(const EigenBase&)
	 */
	template<typename InputType>
	explicit FullPivLU(EigenBase<InputType>& matrix);

	/** Computes the LU decomposition of the given matrix.
	 *
	 * \param matrix the matrix of which to compute the LU decomposition.
	 *               It is required to be nonzero.
	 *
	 * \returns a reference to *this
	 */
	template<typename InputType>
	FullPivLU& compute(const EigenBase<InputType>& matrix)
	{
		m_lu = matrix.derived();
		computeInPlace();
		return *this;
	}

	/** \returns the LU decomposition matrix: the upper-triangular part is U, the
	 * unit-lower-triangular part is L (at least for square matrices; in the non-square
	 * case, special care is needed, see the documentation of class FullPivLU).
	 *
	 * \sa matrixL(), matrixU()
	 */
	inline const MatrixType& matrixLU() const
	{
		eigen_assert(m_isInitialized && "LU is not initialized.");
		return m_lu;
	}

	/** \returns the number of nonzero pivots in the LU decomposition.
	 * Here nonzero is meant in the exact sense, not in a fuzzy sense.
	 * So that notion isn't really intrinsically interesting, but it is
	 * still useful when implementing algorithms.
	 *
	 * \sa rank()
	 */
	inline Index nonzeroPivots() const
	{
		eigen_assert(m_isInitialized && "LU is not initialized.");
		return m_nonzero_pivots;
	}

	/** \returns the absolute value of the biggest pivot, i.e. the biggest
	 *          diagonal coefficient of U.
	 */
	RealScalar maxPivot() const { return m_maxpivot; }

	/** \returns the permutation matrix P
	 *
	 * \sa permutationQ()
	 */
	EIGEN_DEVICE_FUNC inline const PermutationPType& permutationP() const
	{
		eigen_assert(m_isInitialized && "LU is not initialized.");
		return m_p;
	}

	/** \returns the permutation matrix Q
	 *
	 * \sa permutationP()
	 */
	inline const PermutationQType& permutationQ() const
	{
		eigen_assert(m_isInitialized && "LU is not initialized.");
		return m_q;
	}

	/** \returns the kernel of the matrix, also called its null-space. The columns of the returned matrix
	 * will form a basis of the kernel.
	 *
	 * \note If the kernel has dimension zero, then the returned matrix is a column-vector filled with zeros.
	 *
	 * \note This method has to determine which pivots should be considered nonzero.
	 *       For that, it uses the threshold value that you can control by calling
	 *       setThreshold(const RealScalar&).
	 *
	 * Example: \include FullPivLU_kernel.cpp
	 * Output: \verbinclude FullPivLU_kernel.out
	 *
	 * \sa image()
	 */
	inline const internal::kernel_retval<FullPivLU> kernel() const
	{
		eigen_assert(m_isInitialized && "LU is not initialized.");
		return internal::kernel_retval<FullPivLU>(*this);
	}

	/** \returns the image of the matrix, also called its column-space. The columns of the returned matrix
	 * will form a basis of the image (column-space).
	 *
	 * \param originalMatrix the original matrix, of which *this is the LU decomposition.
	 *                       The reason why it is needed to pass it here, is that this allows
	 *                       a large optimization, as otherwise this method would need to reconstruct it
	 *                       from the LU decomposition.
	 *
	 * \note If the image has dimension zero, then the returned matrix is a column-vector filled with zeros.
	 *
	 * \note This method has to determine which pivots should be considered nonzero.
	 *       For that, it uses the threshold value that you can control by calling
	 *       setThreshold(const RealScalar&).
	 *
	 * Example: \include FullPivLU_image.cpp
	 * Output: \verbinclude FullPivLU_image.out
	 *
	 * \sa kernel()
	 */
	inline const internal::image_retval<FullPivLU> image(const MatrixType& originalMatrix) const
	{
		eigen_assert(m_isInitialized && "LU is not initialized.");
		return internal::image_retval<FullPivLU>(*this, originalMatrix);
	}

#ifdef EIGEN_PARSED_BY_DOXYGEN
	/** \return a solution x to the equation Ax=b, where A is the matrix of which
	 * *this is the LU decomposition.
	 *
	 * \param b the right-hand-side of the equation to solve. Can be a vector or a matrix,
	 *          the only requirement in order for the equation to make sense is that
	 *          b.rows()==A.rows(), where A is the matrix of which *this is the LU decomposition.
	 *
	 * \returns a solution.
	 *
	 * \note_about_checking_solutions
	 *
	 * \note_about_arbitrary_choice_of_solution
	 * \note_about_using_kernel_to_study_multiple_solutions
	 *
	 * Example: \include FullPivLU_solve.cpp
	 * Output: \verbinclude FullPivLU_solve.out
	 *
	 * \sa TriangularView::solve(), kernel(), inverse()
	 */
	template<typename Rhs>
	inline const Solve<FullPivLU, Rhs> solve(const MatrixBase<Rhs>& b) const;
#endif

	/** \returns an estimate of the reciprocal condition number of the matrix of which \c *this is
		the LU decomposition.
	  */
	inline RealScalar rcond() const
	{
		eigen_assert(m_isInitialized && "PartialPivLU is not initialized.");
		return internal::rcond_estimate_helper(m_l1_norm, *this);
	}

	/** \returns the determinant of the matrix of which
	 * *this is the LU decomposition. It has only linear complexity
	 * (that is, O(n) where n is the dimension of the square matrix)
	 * as the LU decomposition has already been computed.
	 *
	 * \note This is only for square matrices.
	 *
	 * \note For fixed-size matrices of size up to 4, MatrixBase::determinant() offers
	 *       optimized paths.
	 *
	 * \warning a determinant can be very big or small, so for matrices
	 * of large enough dimension, there is a risk of overflow/underflow.
	 *
	 * \sa MatrixBase::determinant()
	 */
	typename internal::traits<MatrixType>::Scalar determinant() const;

	/** Allows to prescribe a threshold to be used by certain methods, such as rank(),
	 * who need to determine when pivots are to be considered nonzero. This is not used for the
	 * LU decomposition itself.
	 *
	 * When it needs to get the threshold value, Eigen calls threshold(). By default, this
	 * uses a formula to automatically determine a reasonable threshold.
	 * Once you have called the present method setThreshold(const RealScalar&),
	 * your value is used instead.
	 *
	 * \param threshold The new value to use as the threshold.
	 *
	 * A pivot will be considered nonzero if its absolute value is strictly greater than
	 *  \f$ \vert pivot \vert \leqslant threshold \times \vert maxpivot \vert \f$
	 * where maxpivot is the biggest pivot.
	 *
	 * If you want to come back to the default behavior, call setThreshold(Default_t)
	 */
	FullPivLU& setThreshold(const RealScalar& threshold)
	{
		m_usePrescribedThreshold = true;
		m_prescribedThreshold = threshold;
		return *this;
	}

	/** Allows to come back to the default behavior, letting Eigen use its default formula for
	 * determining the threshold.
	 *
	 * You should pass the special object Eigen::Default as parameter here.
	 * \code lu.setThreshold(Eigen::Default); \endcode
	 *
	 * See the documentation of setThreshold(const RealScalar&).
	 */
	FullPivLU& setThreshold(Default_t)
	{
		m_usePrescribedThreshold = false;
		return *this;
	}

	/** Returns the threshold that will be used by certain methods such as rank().
	 *
	 * See the documentation of setThreshold(const RealScalar&).
	 */
	RealScalar threshold() const
	{
		eigen_assert(m_isInitialized || m_usePrescribedThreshold);
		return m_usePrescribedThreshold
				   ? m_prescribedThreshold
				   // this formula comes from experimenting (see "LU precision tuning" thread on the list)
				   // and turns out to be identical to Higham's formula used already in LDLt.
				   : NumTraits<Scalar>::epsilon() * RealScalar(m_lu.diagonalSize());
	}

	/** \returns the rank of the matrix of which *this is the LU decomposition.
	 *
	 * \note This method has to determine which pivots should be considered nonzero.
	 *       For that, it uses the threshold value that you can control by calling
	 *       setThreshold(const RealScalar&).
	 */
	inline Index rank() const
	{
		using std::abs;
		eigen_assert(m_isInitialized && "LU is not initialized.");
		RealScalar premultiplied_threshold = abs(m_maxpivot) * threshold();
		Index result = 0;
		for (Index i = 0; i < m_nonzero_pivots; ++i)
			result += (abs(m_lu.coeff(i, i)) > premultiplied_threshold);
		return result;
	}

	/** \returns the dimension of the kernel of the matrix of which *this is the LU decomposition.
	 *
	 * \note This method has to determine which pivots should be considered nonzero.
	 *       For that, it uses the threshold value that you can control by calling
	 *       setThreshold(const RealScalar&).
	 */
	inline Index dimensionOfKernel() const
	{
		eigen_assert(m_isInitialized && "LU is not initialized.");
		return cols() - rank();
	}

	/** \returns true if the matrix of which *this is the LU decomposition represents an injective
	 *          linear map, i.e. has trivial kernel; false otherwise.
	 *
	 * \note This method has to determine which pivots should be considered nonzero.
	 *       For that, it uses the threshold value that you can control by calling
	 *       setThreshold(const RealScalar&).
	 */
	inline bool isInjective() const
	{
		eigen_assert(m_isInitialized && "LU is not initialized.");
		return rank() == cols();
	}

	/** \returns true if the matrix of which *this is the LU decomposition represents a surjective
	 *          linear map; false otherwise.
	 *
	 * \note This method has to determine which pivots should be considered nonzero.
	 *       For that, it uses the threshold value that you can control by calling
	 *       setThreshold(const RealScalar&).
	 */
	inline bool isSurjective() const
	{
		eigen_assert(m_isInitialized && "LU is not initialized.");
		return rank() == rows();
	}

	/** \returns true if the matrix of which *this is the LU decomposition is invertible.
	 *
	 * \note This method has to determine which pivots should be considered nonzero.
	 *       For that, it uses the threshold value that you can control by calling
	 *       setThreshold(const RealScalar&).
	 */
	inline bool isInvertible() const
	{
		eigen_assert(m_isInitialized && "LU is not initialized.");
		return isInjective() && (m_lu.rows() == m_lu.cols());
	}

	/** \returns the inverse of the matrix of which *this is the LU decomposition.
	 *
	 * \note If this matrix is not invertible, the returned matrix has undefined coefficients.
	 *       Use isInvertible() to first determine whether this matrix is invertible.
	 *
	 * \sa MatrixBase::inverse()
	 */
	inline const Inverse<FullPivLU> inverse() const
	{
		eigen_assert(m_isInitialized && "LU is not initialized.");
		eigen_assert(m_lu.rows() == m_lu.cols() && "You can't take the inverse of a non-square matrix!");
		return Inverse<FullPivLU>(*this);
	}

	MatrixType reconstructedMatrix() const;

	EIGEN_DEVICE_FUNC EIGEN_CONSTEXPR inline Index rows() const EIGEN_NOEXCEPT { return m_lu.rows(); }
	EIGEN_DEVICE_FUNC EIGEN_CONSTEXPR inline Index cols() const EIGEN_NOEXCEPT { return m_lu.cols(); }

#ifndef EIGEN_PARSED_BY_DOXYGEN
	template<typename RhsType, typename DstType>
	void _solve_impl(const RhsType& rhs, DstType& dst) const;

	template<bool Conjugate, typename RhsType, typename DstType>
	void _solve_impl_transposed(const RhsType& rhs, DstType& dst) const;
#endif

  protected:
	static void check_template_parameters() { EIGEN_STATIC_ASSERT_NON_INTEGER(Scalar); }

	void computeInPlace();

	MatrixType m_lu;
	PermutationPType m_p;
	PermutationQType m_q;
	IntColVectorType m_rowsTranspositions;
	IntRowVectorType m_colsTranspositions;
	Index m_nonzero_pivots;
	RealScalar m_l1_norm;
	RealScalar m_maxpivot, m_prescribedThreshold;
	signed char m_det_pq;
	bool m_isInitialized, m_usePrescribedThreshold;
};

template<typename MatrixType>
FullPivLU<MatrixType>::FullPivLU()
	: m_isInitialized(false)
	, m_usePrescribedThreshold(false)
{
}

template<typename MatrixType>
FullPivLU<MatrixType>::FullPivLU(Index rows, Index cols)
	: m_lu(rows, cols)
	, m_p(rows)
	, m_q(cols)
	, m_rowsTranspositions(rows)
	, m_colsTranspositions(cols)
	, m_isInitialized(false)
	, m_usePrescribedThreshold(false)
{
}

template<typename MatrixType>
template<typename InputType>
FullPivLU<MatrixType>::FullPivLU(const EigenBase<InputType>& matrix)
	: m_lu(matrix.rows(), matrix.cols())
	, m_p(matrix.rows())
	, m_q(matrix.cols())
	, m_rowsTranspositions(matrix.rows())
	, m_colsTranspositions(matrix.cols())
	, m_isInitialized(false)
	, m_usePrescribedThreshold(false)
{
	compute(matrix.derived());
}

template<typename MatrixType>
template<typename InputType>
FullPivLU<MatrixType>::FullPivLU(EigenBase<InputType>& matrix)
	: m_lu(matrix.derived())
	, m_p(matrix.rows())
	, m_q(matrix.cols())
	, m_rowsTranspositions(matrix.rows())
	, m_colsTranspositions(matrix.cols())
	, m_isInitialized(false)
	, m_usePrescribedThreshold(false)
{
	computeInPlace();
}

template<typename MatrixType>
void
FullPivLU<MatrixType>::computeInPlace()
{
	check_template_parameters();

	// the permutations are stored as int indices, so just to be sure:
	eigen_assert(m_lu.rows() <= NumTraits<int>::highest() && m_lu.cols() <= NumTraits<int>::highest());

	m_l1_norm = m_lu.cwiseAbs().colwise().sum().maxCoeff();

	const Index size = m_lu.diagonalSize();
	const Index rows = m_lu.rows();
	const Index cols = m_lu.cols();

	// will store the transpositions, before we accumulate them at the end.
	// can't accumulate on-the-fly because that will be done in reverse order for the rows.
	m_rowsTranspositions.resize(m_lu.rows());
	m_colsTranspositions.resize(m_lu.cols());
	Index number_of_transpositions = 0; // number of NONTRIVIAL transpositions, i.e. m_rowsTranspositions[i]!=i

	m_nonzero_pivots = size; // the generic case is that in which all pivots are nonzero (invertible case)
	m_maxpivot = RealScalar(0);

	for (Index k = 0; k < size; ++k) {
		// First, we need to find the pivot.

		// biggest coefficient in the remaining bottom-right corner (starting at row k, col k)
		Index row_of_biggest_in_corner, col_of_biggest_in_corner;
		typedef internal::scalar_score_coeff_op<Scalar> Scoring;
		typedef typename Scoring::result_type Score;
		Score biggest_in_corner;
		biggest_in_corner = m_lu.bottomRightCorner(rows - k, cols - k)
								.unaryExpr(Scoring())
								.maxCoeff(&row_of_biggest_in_corner, &col_of_biggest_in_corner);
		row_of_biggest_in_corner += k; // correct the values! since they were computed in the corner,
		col_of_biggest_in_corner += k; // need to add k to them.

		if (biggest_in_corner == Score(0)) {
			// before exiting, make sure to initialize the still uninitialized transpositions
			// in a sane state without destroying what we already have.
			m_nonzero_pivots = k;
			for (Index i = k; i < size; ++i) {
				m_rowsTranspositions.coeffRef(i) = internal::convert_index<StorageIndex>(i);
				m_colsTranspositions.coeffRef(i) = internal::convert_index<StorageIndex>(i);
			}
			break;
		}

		RealScalar abs_pivot = internal::abs_knowing_score<Scalar>()(
			m_lu(row_of_biggest_in_corner, col_of_biggest_in_corner), biggest_in_corner);
		if (abs_pivot > m_maxpivot)
			m_maxpivot = abs_pivot;

		// Now that we've found the pivot, we need to apply the row/col swaps to
		// bring it to the location (k,k).

		m_rowsTranspositions.coeffRef(k) = internal::convert_index<StorageIndex>(row_of_biggest_in_corner);
		m_colsTranspositions.coeffRef(k) = internal::convert_index<StorageIndex>(col_of_biggest_in_corner);
		if (k != row_of_biggest_in_corner) {
			m_lu.row(k).swap(m_lu.row(row_of_biggest_in_corner));
			++number_of_transpositions;
		}
		if (k != col_of_biggest_in_corner) {
			m_lu.col(k).swap(m_lu.col(col_of_biggest_in_corner));
			++number_of_transpositions;
		}

		// Now that the pivot is at the right location, we update the remaining
		// bottom-right corner by Gaussian elimination.

		if (k < rows - 1)
			m_lu.col(k).tail(rows - k - 1) /= m_lu.coeff(k, k);
		if (k < size - 1)
			m_lu.block(k + 1, k + 1, rows - k - 1, cols - k - 1).noalias() -=
				m_lu.col(k).tail(rows - k - 1) * m_lu.row(k).tail(cols - k - 1);
	}

	// the main loop is over, we still have to accumulate the transpositions to find the
	// permutations P and Q

	m_p.setIdentity(rows);
	for (Index k = size - 1; k >= 0; --k)
		m_p.applyTranspositionOnTheRight(k, m_rowsTranspositions.coeff(k));

	m_q.setIdentity(cols);
	for (Index k = 0; k < size; ++k)
		m_q.applyTranspositionOnTheRight(k, m_colsTranspositions.coeff(k));

	m_det_pq = (number_of_transpositions % 2) ? -1 : 1;

	m_isInitialized = true;
}

template<typename MatrixType>
typename internal::traits<MatrixType>::Scalar
FullPivLU<MatrixType>::determinant() const
{
	eigen_assert(m_isInitialized && "LU is not initialized.");
	eigen_assert(m_lu.rows() == m_lu.cols() && "You can't take the determinant of a non-square matrix!");
	return Scalar(m_det_pq) * Scalar(m_lu.diagonal().prod());
}

/** \returns the matrix represented by the decomposition,
 * i.e., it returns the product: \f$ P^{-1} L U Q^{-1} \f$.
 * This function is provided for debug purposes. */
template<typename MatrixType>
MatrixType
FullPivLU<MatrixType>::reconstructedMatrix() const
{
	eigen_assert(m_isInitialized && "LU is not initialized.");
	const Index smalldim = (std::min)(m_lu.rows(), m_lu.cols());
	// LU
	MatrixType res(m_lu.rows(), m_lu.cols());
	// FIXME the .toDenseMatrix() should not be needed...
	res = m_lu.leftCols(smalldim).template triangularView<UnitLower>().toDenseMatrix() *
		  m_lu.topRows(smalldim).template triangularView<Upper>().toDenseMatrix();

	// P^{-1}(LU)
	res = m_p.inverse() * res;

	// (P^{-1}LU)Q^{-1}
	res = res * m_q.inverse();

	return res;
}

/********* Implementation of kernel() **************************************************/

namespace internal {
template<typename _MatrixType>
struct kernel_retval<FullPivLU<_MatrixType>> : kernel_retval_base<FullPivLU<_MatrixType>>
{
	EIGEN_MAKE_KERNEL_HELPERS(FullPivLU<_MatrixType>)

	enum
	{
		MaxSmallDimAtCompileTime =
			EIGEN_SIZE_MIN_PREFER_FIXED(MatrixType::MaxColsAtCompileTime, MatrixType::MaxRowsAtCompileTime)
	};

	template<typename Dest>
	void evalTo(Dest& dst) const
	{
		using std::abs;
		const Index cols = dec().matrixLU().cols(), dimker = cols - rank();
		if (dimker == 0) {
			// The Kernel is just {0}, so it doesn't have a basis properly speaking, but let's
			// avoid crashing/asserting as that depends on floating point calculations. Let's
			// just return a single column vector filled with zeros.
			dst.setZero();
			return;
		}

		/* Let us use the following lemma:
		 *
		 * Lemma: If the matrix A has the LU decomposition PAQ = LU,
		 * then Ker A = Q(Ker U).
		 *
		 * Proof: trivial: just keep in mind that P, Q, L are invertible.
		 */

		/* Thus, all we need to do is to compute Ker U, and then apply Q.
		 *
		 * U is upper triangular, with eigenvalues sorted so that any zeros appear at the end.
		 * Thus, the diagonal of U ends with exactly
		 * dimKer zero's. Let us use that to construct dimKer linearly
		 * independent vectors in Ker U.
		 */

		Matrix<Index, Dynamic, 1, 0, MaxSmallDimAtCompileTime, 1> pivots(rank());
		RealScalar premultiplied_threshold = dec().maxPivot() * dec().threshold();
		Index p = 0;
		for (Index i = 0; i < dec().nonzeroPivots(); ++i)
			if (abs(dec().matrixLU().coeff(i, i)) > premultiplied_threshold)
				pivots.coeffRef(p++) = i;
		eigen_internal_assert(p == rank());

		// we construct a temporaty trapezoid matrix m, by taking the U matrix and
		// permuting the rows and cols to bring the nonnegligible pivots to the top of
		// the main diagonal. We need that to be able to apply our triangular solvers.
		// FIXME when we get triangularView-for-rectangular-matrices, this can be simplified
		Matrix<typename MatrixType::Scalar,
			   Dynamic,
			   Dynamic,
			   MatrixType::Options,
			   MaxSmallDimAtCompileTime,
			   MatrixType::MaxColsAtCompileTime>
			m(dec().matrixLU().block(0, 0, rank(), cols));
		for (Index i = 0; i < rank(); ++i) {
			if (i)
				m.row(i).head(i).setZero();
			m.row(i).tail(cols - i) = dec().matrixLU().row(pivots.coeff(i)).tail(cols - i);
		}
		m.block(0, 0, rank(), rank());
		m.block(0, 0, rank(), rank()).template triangularView<StrictlyLower>().setZero();
		for (Index i = 0; i < rank(); ++i)
			m.col(i).swap(m.col(pivots.coeff(i)));

		// ok, we have our trapezoid matrix, we can apply the triangular solver.
		// notice that the math behind this suggests that we should apply this to the
		// negative of the RHS, but for performance we just put the negative sign elsewhere, see below.
		m.topLeftCorner(rank(), rank()).template triangularView<Upper>().solveInPlace(m.topRightCorner(rank(), dimker));

		// now we must undo the column permutation that we had applied!
		for (Index i = rank() - 1; i >= 0; --i)
			m.col(i).swap(m.col(pivots.coeff(i)));

		// see the negative sign in the next line, that's what we were talking about above.
		for (Index i = 0; i < rank(); ++i)
			dst.row(dec().permutationQ().indices().coeff(i)) = -m.row(i).tail(dimker);
		for (Index i = rank(); i < cols; ++i)
			dst.row(dec().permutationQ().indices().coeff(i)).setZero();
		for (Index k = 0; k < dimker; ++k)
			dst.coeffRef(dec().permutationQ().indices().coeff(rank() + k), k) = Scalar(1);
	}
};

/***** Implementation of image() *****************************************************/

template<typename _MatrixType>
struct image_retval<FullPivLU<_MatrixType>> : image_retval_base<FullPivLU<_MatrixType>>
{
	EIGEN_MAKE_IMAGE_HELPERS(FullPivLU<_MatrixType>)

	enum
	{
		MaxSmallDimAtCompileTime =
			EIGEN_SIZE_MIN_PREFER_FIXED(MatrixType::MaxColsAtCompileTime, MatrixType::MaxRowsAtCompileTime)
	};

	template<typename Dest>
	void evalTo(Dest& dst) const
	{
		using std::abs;
		if (rank() == 0) {
			// The Image is just {0}, so it doesn't have a basis properly speaking, but let's
			// avoid crashing/asserting as that depends on floating point calculations. Let's
			// just return a single column vector filled with zeros.
			dst.setZero();
			return;
		}

		Matrix<Index, Dynamic, 1, 0, MaxSmallDimAtCompileTime, 1> pivots(rank());
		RealScalar premultiplied_threshold = dec().maxPivot() * dec().threshold();
		Index p = 0;
		for (Index i = 0; i < dec().nonzeroPivots(); ++i)
			if (abs(dec().matrixLU().coeff(i, i)) > premultiplied_threshold)
				pivots.coeffRef(p++) = i;
		eigen_internal_assert(p == rank());

		for (Index i = 0; i < rank(); ++i)
			dst.col(i) = originalMatrix().col(dec().permutationQ().indices().coeff(pivots.coeff(i)));
	}
};

/***** Implementation of solve() *****************************************************/

} // end namespace internal

#ifndef EIGEN_PARSED_BY_DOXYGEN
template<typename _MatrixType>
template<typename RhsType, typename DstType>
void
FullPivLU<_MatrixType>::_solve_impl(const RhsType& rhs, DstType& dst) const
{
	/* The decomposition PAQ = LU can be rewritten as A = P^{-1} L U Q^{-1}.
	 * So we proceed as follows:
	 * Step 1: compute c = P * rhs.
	 * Step 2: replace c by the solution x to Lx = c. Exists because L is invertible.
	 * Step 3: replace c by the solution x to Ux = c. May or may not exist.
	 * Step 4: result = Q * c;
	 */

	const Index rows = this->rows(), cols = this->cols(), nonzero_pivots = this->rank();
	const Index smalldim = (std::min)(rows, cols);

	if (nonzero_pivots == 0) {
		dst.setZero();
		return;
	}

	typename RhsType::PlainObject c(rhs.rows(), rhs.cols());

	// Step 1
	c = permutationP() * rhs;

	// Step 2
	m_lu.topLeftCorner(smalldim, smalldim).template triangularView<UnitLower>().solveInPlace(c.topRows(smalldim));
	if (rows > cols)
		c.bottomRows(rows - cols) -= m_lu.bottomRows(rows - cols) * c.topRows(cols);

	// Step 3
	m_lu.topLeftCorner(nonzero_pivots, nonzero_pivots)
		.template triangularView<Upper>()
		.solveInPlace(c.topRows(nonzero_pivots));

	// Step 4
	for (Index i = 0; i < nonzero_pivots; ++i)
		dst.row(permutationQ().indices().coeff(i)) = c.row(i);
	for (Index i = nonzero_pivots; i < m_lu.cols(); ++i)
		dst.row(permutationQ().indices().coeff(i)).setZero();
}

template<typename _MatrixType>
template<bool Conjugate, typename RhsType, typename DstType>
void
FullPivLU<_MatrixType>::_solve_impl_transposed(const RhsType& rhs, DstType& dst) const
{
	/* The decomposition PAQ = LU can be rewritten as A = P^{-1} L U Q^{-1},
	 * and since permutations are real and unitary, we can write this
	 * as   A^T = Q U^T L^T P,
	 * So we proceed as follows:
	 * Step 1: compute c = Q^T rhs.
	 * Step 2: replace c by the solution x to U^T x = c. May or may not exist.
	 * Step 3: replace c by the solution x to L^T x = c.
	 * Step 4: result = P^T c.
	 * If Conjugate is true, replace "^T" by "^*" above.
	 */

	const Index rows = this->rows(), cols = this->cols(), nonzero_pivots = this->rank();
	const Index smalldim = (std::min)(rows, cols);

	if (nonzero_pivots == 0) {
		dst.setZero();
		return;
	}

	typename RhsType::PlainObject c(rhs.rows(), rhs.cols());

	// Step 1
	c = permutationQ().inverse() * rhs;

	// Step 2
	m_lu.topLeftCorner(nonzero_pivots, nonzero_pivots)
		.template triangularView<Upper>()
		.transpose()
		.template conjugateIf<Conjugate>()
		.solveInPlace(c.topRows(nonzero_pivots));

	// Step 3
	m_lu.topLeftCorner(smalldim, smalldim)
		.template triangularView<UnitLower>()
		.transpose()
		.template conjugateIf<Conjugate>()
		.solveInPlace(c.topRows(smalldim));

	// Step 4
	PermutationPType invp = permutationP().inverse().eval();
	for (Index i = 0; i < smalldim; ++i)
		dst.row(invp.indices().coeff(i)) = c.row(i);
	for (Index i = smalldim; i < rows; ++i)
		dst.row(invp.indices().coeff(i)).setZero();
}

#endif

namespace internal {

/***** Implementation of inverse() *****************************************************/
template<typename DstXprType, typename MatrixType>
struct Assignment<DstXprType,
				  Inverse<FullPivLU<MatrixType>>,
				  internal::assign_op<typename DstXprType::Scalar, typename FullPivLU<MatrixType>::Scalar>,
				  Dense2Dense>
{
	typedef FullPivLU<MatrixType> LuType;
	typedef Inverse<LuType> SrcXprType;
	static void run(DstXprType& dst,
					const SrcXprType& src,
					const internal::assign_op<typename DstXprType::Scalar, typename MatrixType::Scalar>&)
	{
		dst = src.nestedExpression().solve(MatrixType::Identity(src.rows(), src.cols()));
	}
};
} // end namespace internal

/******* MatrixBase methods *****************************************************************/

/** \lu_module
 *
 * \return the full-pivoting LU decomposition of \c *this.
 *
 * \sa class FullPivLU
 */
template<typename Derived>
inline const FullPivLU<typename MatrixBase<Derived>::PlainObject>
MatrixBase<Derived>::fullPivLu() const
{
	return FullPivLU<PlainObject>(eval());
}

} // end namespace Eigen

#endif // EIGEN_LU_H
